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#@local POWERMODINT_GAP,b,bigPos,bigNeg,checkPValuationInt,data,dataHex
#@local dataInv,dataNonZero,e,f,g,i,k,m,mysource,pow,r,smlNeg,smlPos,x,y
#@local naivQM,ps,checkROOT_INT,P,a,n,p,x1,x2
gap> START_TEST("intarith.tst");
gap> 1 + 1;
2
gap> 1 - 1;
0
gap> 1 * 1;
1
gap> 1 / 1;
1
gap> RemInt(1,1);
0
gap> QuoInt(1,1);
1
#
# We need to cover four cases for (opL, opR) of each binary operation:
# (small,small), (small,gmp), (gmp,small), (gmp,gmp). In addition, due
# to the way GAP represents negative values internally, in many cases we
# also need to check positive and negative values separately, both small
# and gmp. We choose values with nice decimal representation, to make it
# easier to manually verify the results.
#
gap> smlPos := 10000; bigPos := 100000000000000000000;
10000
100000000000000000000
gap> smlNeg := -smlPos; bigNeg := -bigPos;
-10000
-100000000000000000000
gap> data := [ bigNeg-1, bigNeg, smlNeg, -1, 0, 1,smlPos, bigPos, bigPos+1 ];
[ -100000000000000000001, -100000000000000000000, -10000, -1, 0, 1, 10000,
100000000000000000000, 100000000000000000001 ]
gap> dataNonZero := [ bigNeg-1, bigNeg, smlNeg, -1, 1,smlPos, bigPos, bigPos+1 ];
[ -100000000000000000001, -100000000000000000000, -10000, -1, 1, 10000,
100000000000000000000, 100000000000000000001 ]
#
# LtInt
#
gap> List(data, x -> bigNeg < x);
[ false, false, true, true, true, true, true, true, true ]
gap> List(data, x -> smlNeg < x);
[ false, false, false, true, true, true, true, true, true ]
gap> List(data, x -> smlPos < x);
[ false, false, false, false, false, false, false, true, true ]
gap> List(data, x -> bigPos < x);
[ false, false, false, false, false, false, false, false, true ]
#
gap> List(data, x -> x < bigNeg);
[ true, false, false, false, false, false, false, false, false ]
gap> List(data, x -> x < smlNeg);
[ true, true, false, false, false, false, false, false, false ]
gap> List(data, x -> x < smlPos);
[ true, true, true, true, true, true, false, false, false ]
gap> List(data, x -> x < bigPos);
[ true, true, true, true, true, true, true, false, false ]
#
# EqInt
#
gap> List(data, x -> bigNeg = x);
[ false, true, false, false, false, false, false, false, false ]
gap> List(data, x -> smlNeg = x);
[ false, false, true, false, false, false, false, false, false ]
gap> List(data, x -> smlPos = x);
[ false, false, false, false, false, false, true, false, false ]
gap> List(data, x -> bigPos = x);
[ false, false, false, false, false, false, false, true, false ]
# Check for < on equal, but not identical large numbers
# See issue #3474
gap> x1 := 2^80;; x2 := 2^80;;
gap> x1 < x2;
false
gap> x1 = x2;
true
gap> -x1 < -x2;
false
gap> -x1 = -x2;
true
# check symmetry
gap> ForAll(data, x -> ForAll(data, y -> (x=y) = (y=x)));
true
#
# SumInt
#
gap> List(data, x -> bigNeg + x);
[ -200000000000000000001, -200000000000000000000, -100000000000000010000,
-100000000000000000001, -100000000000000000000, -99999999999999999999,
-99999999999999990000, 0, 1 ]
gap> List(data, x -> smlNeg + x);
[ -100000000000000010001, -100000000000000010000, -20000, -10001, -10000,
-9999, 0, 99999999999999990000, 99999999999999990001 ]
gap> List(data, x -> smlPos + x);
[ -99999999999999990001, -99999999999999990000, 0, 9999, 10000, 10001, 20000,
100000000000000010000, 100000000000000010001 ]
gap> List(data, x -> bigPos + x);
[ -1, 0, 99999999999999990000, 99999999999999999999, 100000000000000000000,
100000000000000000001, 100000000000000010000, 200000000000000000000,
200000000000000000001 ]
# check commutativity
gap> ForAll(data, x -> ForAll(data, y -> (x+y) = (y+x)));
true
#
# DiffInt
#
gap> List(data, x -> bigNeg - x);
[ 1, 0, -99999999999999990000, -99999999999999999999, -100000000000000000000,
-100000000000000000001, -100000000000000010000, -200000000000000000000,
-200000000000000000001 ]
gap> List(data, x -> smlNeg - x);
[ 99999999999999990001, 99999999999999990000, 0, -9999, -10000, -10001,
-20000, -100000000000000010000, -100000000000000010001 ]
gap> List(data, x -> smlPos - x);
[ 100000000000000010001, 100000000000000010000, 20000, 10001, 10000, 9999, 0,
-99999999999999990000, -99999999999999990001 ]
gap> List(data, x -> bigPos - x);
[ 200000000000000000001, 200000000000000000000, 100000000000000010000,
100000000000000000001, 100000000000000000000, 99999999999999999999,
99999999999999990000, 0, -1 ]
# check anti-commutativity
gap> ForAll(data, x -> ForAll(data, y -> (x-y) = -(y-x)));
true
#
# ProdInt
#
gap> List(data, x -> bigNeg * x);
[ 10000000000000000000100000000000000000000,
10000000000000000000000000000000000000000, 1000000000000000000000000,
100000000000000000000, 0, -100000000000000000000,
-1000000000000000000000000, -10000000000000000000000000000000000000000,
-10000000000000000000100000000000000000000 ]
gap> List(data, x -> smlNeg * x);
[ 1000000000000000000010000, 1000000000000000000000000, 100000000, 10000, 0,
-10000, -100000000, -1000000000000000000000000, -1000000000000000000010000 ]
gap> List(data, x -> smlPos * x);
[ -1000000000000000000010000, -1000000000000000000000000, -100000000, -10000,
0, 10000, 100000000, 1000000000000000000000000, 1000000000000000000010000 ]
gap> List(data, x -> bigPos * x);
[ -10000000000000000000100000000000000000000,
-10000000000000000000000000000000000000000, -1000000000000000000000000,
-100000000000000000000, 0, 100000000000000000000, 1000000000000000000000000,
10000000000000000000000000000000000000000,
10000000000000000000100000000000000000000 ]
# check commutativity
gap> ForAll(data, x -> ForAll(data, y -> (x*y) = (y*x)));
true
#
# ProdIntObj
#
gap> 0 * 1.;
0.
gap> 1 * 1.;
1.
gap> 2 * 1.;
2.
gap> -1 * 1.;
-1.
gap> -2 * 1.;
-2.
gap> 2^60 * 1.;
1.15292e+18
gap> -2^60 * 1.;
-1.15292e+18
#
# PowInt
#
gap> List(data{[5..9]}, x -> 0^x);
[ 1, 0, 0, 0, 0 ]
gap> 0^-1;
Error, Integer operands: <base> must not be zero
gap> 0^-12345678901234567890;
Error, Integer operands: <base> must not be zero
gap> List(data, x -> 1^x);
[ 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
gap> List(data, x -> (-1)^x);
[ -1, 1, 1, -1, 1, -1, 1, 1, -1 ]
#
gap> List(data, x -> x^0);
[ 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
gap> ForAll(data, x -> x^1 = x);
true
gap> ForAll(data, x -> x^2 = x*x);
true
gap> ForAll(data, x -> x^3 = x*x*x);
true
gap> 2^100;
1267650600228229401496703205376
#
# PowObjInt
#
gap> [[1]] ^ smlNeg;
[ [ 1 ] ]
gap> [[1]] ^ bigNeg;
[ [ 1 ] ]
gap> [[0]] ^ smlNeg;
Error, Operations: <obj> must have an inverse
gap> [[0]] ^ bigNeg;
Error, Operations: <obj> must have an inverse
#
# ModInt
#
gap> List(dataNonZero, x -> bigNeg mod x);
[ 1, 0, 0, 0, 0, 0, 0, 1 ]
gap> List(dataNonZero, x -> smlNeg mod x);
[ 99999999999999990001, 99999999999999990000, 0, 0, 0, 0,
99999999999999990000, 99999999999999990001 ]
gap> List(dataNonZero, x -> smlPos mod x);
[ 10000, 10000, 0, 0, 0, 0, 10000, 10000 ]
gap> List(dataNonZero, x -> bigPos mod x);
[ 100000000000000000000, 0, 0, 0, 0, 0, 0, 100000000000000000000 ]
#
gap> List(data, x -> x mod bigNeg);
[ 99999999999999999999, 0, 99999999999999990000, 99999999999999999999, 0, 1,
10000, 0, 1 ]
gap> List(data, x -> x mod smlNeg);
[ 9999, 0, 0, 9999, 0, 1, 0, 0, 1 ]
gap> List(data, x -> x mod smlPos);
[ 9999, 0, 0, 9999, 0, 1, 0, 0, 1 ]
gap> List(data, x -> x mod bigPos);
[ 99999999999999999999, 0, 99999999999999990000, 99999999999999999999, 0, 1,
10000, 0, 1 ]
#
gap> bigNeg mod 0;
Error, Integer operations: <divisor> must be a nonzero integer (not the intege\
r 0)
gap> smlNeg mod 0;
Error, Integer operations: <divisor> must be a nonzero integer (not the intege\
r 0)
gap> smlPos mod 0;
Error, Integer operations: <divisor> must be a nonzero integer (not the intege\
r 0)
gap> bigPos mod 0;
Error, Integer operations: <divisor> must be a nonzero integer (not the intege\
r 0)
# test optimization for moduli which are a power of 2
gap> x:=2^80;
1208925819614629174706176
gap> ForAll([1..80], i -> x mod 2^i = 0);
true
gap> ForAll([1..80], i -> x mod -2^i = 0);
true
gap> ForAll([1..80], i -> (-x) mod 2^i = 0);
true
gap> ForAll([1..80], i -> (-x) mod -2^i = 0);
true
#
gap> y:=2^80-1;
1208925819614629174706175
gap> ForAll([1..80], i -> y mod 2^i = 2^i-1);
true
gap> ForAll([1..80], i -> y mod -2^i = 2^i-1);
true
gap> ForAll([1..80], i -> (-y) mod 2^i = 1);
true
gap> ForAll([1..80], i -> (-y) mod -2^i = 1);
true
# corner cases
gap> (-2^28) mod (2^28);
0
gap> (-2^60) mod (2^60);
0
# check the optimization for two large integers, one with more limbs than the other
gap> bigPos mod (bigPos*bigPos);
100000000000000000000
gap> bigNeg mod (bigPos*bigPos);
9999999999999999999900000000000000000000
gap> bigPos mod (bigPos*bigNeg);
100000000000000000000
gap> bigNeg mod (bigPos*bigNeg);
9999999999999999999900000000000000000000
#
# QuoInt
#
gap> List(dataNonZero, x -> QuoInt(bigNeg, x));
[ 0, 1, 10000000000000000, 100000000000000000000, -100000000000000000000,
-10000000000000000, -1, 0 ]
gap> List(dataNonZero, x -> QuoInt(smlNeg, x));
[ 0, 0, 1, 10000, -10000, -1, 0, 0 ]
gap> List(dataNonZero, x -> QuoInt(smlPos, x));
[ 0, 0, -1, -10000, 10000, 1, 0, 0 ]
gap> List(dataNonZero, x -> QuoInt(bigPos, x));
[ 0, -1, -10000000000000000, -100000000000000000000, 100000000000000000000,
10000000000000000, 1, 0 ]
#
gap> List(data, x -> QuoInt(x, bigNeg));
[ 1, 1, 0, 0, 0, 0, 0, -1, -1 ]
gap> List(data, x -> QuoInt(x, smlNeg));
[ 10000000000000000, 10000000000000000, 1, 0, 0, 0, -1, -10000000000000000,
-10000000000000000 ]
gap> List(data, x -> QuoInt(x, smlPos));
[ -10000000000000000, -10000000000000000, -1, 0, 0, 0, 1, 10000000000000000,
10000000000000000 ]
gap> List(data, x -> QuoInt(x, bigPos));
[ -1, -1, 0, 0, 0, 0, 0, 1, 1 ]
#
gap> QuoInt(bigNeg, 0);
Error, Integer operations: <divisor> must be a nonzero integer (not the intege\
r 0)
gap> QuoInt(smlNeg, 0);
Error, Integer operations: <divisor> must be a nonzero integer (not the intege\
r 0)
gap> QuoInt(smlPos, 0);
Error, Integer operations: <divisor> must be a nonzero integer (not the intege\
r 0)
gap> QuoInt(bigPos, 0);
Error, Integer operations: <divisor> must be a nonzero integer (not the intege\
r 0)
gap> QuoInt(fail,1);
Error, QUO_INT: <a> must be an integer (not the value 'fail')
gap> QuoInt(1,fail);
Error, QUO_INT: <b> must be an integer (not the value 'fail')
# corner cases
gap> QuoInt(-2^28, -1);
268435456
gap> QuoInt(-2^60, -1);
1152921504606846976
gap> QuoInt(-2^28, 2^28);
-1
gap> QuoInt(-2^60, 2^60);
-1
# check the optimization for two large integers, one with more limbs than the other
gap> QuoInt(bigPos, bigPos*bigPos);
0
#
# RemInt
#
gap> List(dataNonZero, x -> RemInt(bigNeg, x));
[ -100000000000000000000, 0, 0, 0, 0, 0, 0, -100000000000000000000 ]
gap> List(dataNonZero, x -> RemInt(smlNeg, x));
[ -10000, -10000, 0, 0, 0, 0, -10000, -10000 ]
gap> List(dataNonZero, x -> RemInt(smlPos, x));
[ 10000, 10000, 0, 0, 0, 0, 10000, 10000 ]
gap> List(dataNonZero, x -> RemInt(bigPos, x));
[ 100000000000000000000, 0, 0, 0, 0, 0, 0, 100000000000000000000 ]
#
gap> List(data, x -> RemInt(x, bigNeg));
[ -1, 0, -10000, -1, 0, 1, 10000, 0, 1 ]
gap> List(data, x -> RemInt(x, smlNeg));
[ -1, 0, 0, -1, 0, 1, 0, 0, 1 ]
gap> List(data, x -> RemInt(x, smlPos));
[ -1, 0, 0, -1, 0, 1, 0, 0, 1 ]
gap> List(data, x -> RemInt(x, bigPos));
[ -1, 0, -10000, -1, 0, 1, 10000, 0, 1 ]
#
gap> RemInt(bigNeg, 0);
Error, Integer operations: <divisor> must be a nonzero integer (not the intege\
r 0)
gap> RemInt(smlNeg, 0);
Error, Integer operations: <divisor> must be a nonzero integer (not the intege\
r 0)
gap> RemInt(smlPos, 0);
Error, Integer operations: <divisor> must be a nonzero integer (not the intege\
r 0)
gap> RemInt(bigPos, 0);
Error, Integer operations: <divisor> must be a nonzero integer (not the intege\
r 0)
gap> RemInt(fail,1);
Error, REM_INT: <a> must be an integer (not the value 'fail')
gap> RemInt(1,fail);
Error, REM_INT: <b> must be an integer (not the value 'fail')
# corner cases
gap> RemInt(-2^28, 2^28);
0
gap> RemInt(-2^60, 2^60);
0
# test optimization for moduli which are a power of 2
gap> x:=2^80;
1208925819614629174706176
gap> ForAll([1..80], i -> RemInt(x, 2^i) = 0);
true
gap> ForAll([1..80], i -> RemInt(x, -2^i) = 0);
true
gap> ForAll([1..80], i -> RemInt(-x, 2^i) = 0);
true
gap> ForAll([1..80], i -> RemInt(-x, -2^i) = 0);
true
#
gap> y:=2^80-1;
1208925819614629174706175
gap> ForAll([1..80], i -> RemInt(y, 2^i) = 2^i-1);
true
gap> ForAll([1..80], i -> RemInt(y, -2^i) = 2^i-1);
true
gap> ForAll([1..80], i -> RemInt(-y, 2^i) = -2^i+1);
true
gap> ForAll([1..80], i -> RemInt(-y, -2^i) = -2^i+1);
true
# check the optimization for two large integers, one with more limbs than the other
gap> RemInt(bigPos, bigPos*bigPos);
100000000000000000000
gap> RemInt(bigNeg, bigPos*bigPos);
-100000000000000000000
gap> RemInt(bigPos, bigPos*bigNeg);
100000000000000000000
gap> RemInt(bigNeg, bigPos*bigNeg);
-100000000000000000000
#
# GcdInt
#
gap> List(data, x -> GcdInt(bigNeg, x));
[ 1, 100000000000000000000, 10000, 1, 100000000000000000000, 1, 10000,
100000000000000000000, 1 ]
gap> List(data, x -> GcdInt(smlNeg, x));
[ 1, 10000, 10000, 1, 10000, 1, 10000, 10000, 1 ]
gap> List(data, x -> GcdInt(smlPos, x));
[ 1, 10000, 10000, 1, 10000, 1, 10000, 10000, 1 ]
gap> List(data, x -> GcdInt(bigPos, x));
[ 1, 100000000000000000000, 10000, 1, 100000000000000000000, 1, 10000,
100000000000000000000, 1 ]
#
gap> g:=3^10*257;; a:=g*991;; b:=g*1601;;
gap> GcdInt(2^20*a, 2^30*b) = 2^20*g;
true
gap> GcdInt(2^80*a, 2^40*b) = 2^40*g;
true
gap> GcdInt(2^80*a, 2^90*b) = 2^80*g;
true
# check symmetry
gap> ForAll(data, x -> ForAll(data, y -> GcdInt(x,y) = GcdInt(y,x)));
true
#
gap> GcdInt(fail,1);
Error, GCD_INT: <a> must be an integer (not the value 'fail')
gap> GcdInt(1,fail);
Error, GCD_INT: <b> must be an integer (not the value 'fail')
# corner cases
gap> GcdInt(0, 0);
0
gap> GcdInt(-2^60, -2^60);
1152921504606846976
gap> GcdInt(-2^28, -2^28);
268435456
#
# LcmInt
#
gap> List(data, x -> LcmInt(bigNeg, x));
[ 10000000000000000000100000000000000000000, 100000000000000000000,
100000000000000000000, 100000000000000000000, 0, 100000000000000000000,
100000000000000000000, 100000000000000000000,
10000000000000000000100000000000000000000 ]
gap> List(data, x -> LcmInt(smlNeg, x));
[ 1000000000000000000010000, 100000000000000000000, 10000, 10000, 0, 10000,
10000, 100000000000000000000, 1000000000000000000010000 ]
gap> List(data, x -> LcmInt(smlPos, x));
[ 1000000000000000000010000, 100000000000000000000, 10000, 10000, 0, 10000,
10000, 100000000000000000000, 1000000000000000000010000 ]
gap> List(data, x -> LcmInt(bigPos, x));
[ 10000000000000000000100000000000000000000, 100000000000000000000,
100000000000000000000, 100000000000000000000, 0, 100000000000000000000,
100000000000000000000, 100000000000000000000,
10000000000000000000100000000000000000000 ]
#
gap> g:=3^10*257;; a:=g*991;; b:=g*1601;;
gap> LcmInt(2^20*a, 2^30*b) = 2^30*3^10*257*991*1601;
true
gap> LcmInt(2^80*a, 2^40*b) = 2^80*3^10*257*991*1601;
true
gap> LcmInt(2^80*a, 2^90*b) = 2^90*3^10*257*991*1601;
true
# check symmetry
gap> ForAll(data, x -> ForAll(data, y -> LcmInt(x,y) = LcmInt(y,x)));
true
#
gap> LcmInt(fail,1);
Error, LcmInt: <a> must be an integer (not the value 'fail')
gap> LcmInt(1,fail);
Error, LcmInt: <b> must be an integer (not the value 'fail')
# corner cases
gap> LcmInt(0, 0);
0
gap> LcmInt(-2^60, -2^60);
1152921504606846976
gap> LcmInt(-2^28, -2^28);
268435456
#
# AInvInt
#
gap> dataInv := List(data, x -> -x);
[ 100000000000000000001, 100000000000000000000, 10000, 1, 0, -1, -10000,
-100000000000000000000, -100000000000000000001 ]
gap> dataInv + data;
[ 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
#
# AbsInt
#
gap> List(data, AbsInt);
[ 100000000000000000001, 100000000000000000000, 10000, 1, 0, 1, 10000,
100000000000000000000, 100000000000000000001 ]
gap> AbsInt(-2^28) = 2^28; # corner case on 32bit systems
true
gap> AbsInt(-2^60) = 2^60; # corner case on 64bit systems
true
gap> AbsInt(fail);
Error, AbsInt: <op> must be a rational (not the value 'fail')
gap> ABS_INT(fail);
Error, ABS_INT: <n> must be an integer (not the value 'fail')
#
# SignInt
#
gap> List(data, SignInt);
[ -1, -1, -1, -1, 0, 1, 1, 1, 1 ]
gap> SignInt(fail);
Error, SignInt: <op> must be a rational (not the value 'fail')
gap> SIGN_INT(fail);
Error, SIGN_INT: <n> must be an integer (not the value 'fail')
#
# QuotientMod
#
# test "... q is either zero (if r is divisible by m) ... "
gap> ForAll([1..12], m -> ForAll(m*[-m..m], r -> QuotientMod(0,r,m)=0));
true
# test that 0 divides nothing other than multiples of m
gap> ForAll([1..10], m -> ForAll([1..m-1], r -> QuotientMod(r,0,m)=fail));
true
# brute force test a bunch of inputs
gap> naivQM:={r,s,m}->First([0..m-1], q -> ((q*s-r) mod m) = 0);;
gap> f := m -> SetX([-2*m..2*m], [-2*m..2*m], {r,s} -> QuotientMod(r,s,m) = naivQM(r,s,m));;
gap> Set([1..12], f);
[ [ true ] ]
# test that m = 0 is forbidden
gap> QuotientMod(5, 4, 0);
Error, Integer operations: <divisor> must be a nonzero integer (not the intege\
r 0)
# some old test cases, to show case that issue #149 is resolved
gap> QuotientMod(2, 4, 6);
2
gap> QuotientMod(4, 2, 6);
2
gap> QuotientMod(-2, 2, 6);
2
gap> QuotientMod(4, 8, 6);
2
#
# HexStringInt and IntHexString
#
gap> dataHex := List(data, HexStringInt);
[ "-56BC75E2D63100001", "-56BC75E2D63100000", "-2710", "-1", "0", "1",
"2710", "56BC75E2D63100000", "56BC75E2D63100001" ]
gap> HexStringInt("abc");
Error, HexStringInt: <n> must be an integer (not a list (string))
gap> List(dataHex, IntHexString) = data;
true
gap> dataHex; # HexStringInt used to destroy its argument
[ "-56BC75E2D63100001", "-56BC75E2D63100000", "-2710", "-1", "0", "1",
"2710", "56BC75E2D63100000", "56BC75E2D63100001" ]
gap> IntHexString("");
0
gap> IntHexString("a");
10
gap> IntHexString("A");
10
gap> IntHexString("x");
Error, IntHexString: invalid character in hex-string
gap> IntHexString("56BC75E2D63100000x");
Error, IntHexString: invalid character in hex-string
gap> IntHexString("56BC75E2D63100000@");
Error, IntHexString: invalid character in hex-string
gap> IntHexString(0);
Error, IntHexString: <str> must be a string (not the integer 0)
gap> IntHexString("-1000000000000000");
-1152921504606846976
#
# Log2Int
#
gap> List(data, Log2Int);
[ 66, 66, 13, 0, -1, 0, 13, 66, 66 ]
gap> Log2Int("abc");
Error, Log2Int: <n> must be an integer (not a list (string))
gap> List([-5..5], Log2Int);
[ 2, 2, 1, 1, 0, -1, 0, 1, 1, 2, 2 ]
gap> ForAll([2..100], x -> Log2Int(2^x) = x and
> Log2Int(2^x-1) = x-1 and
> Log2Int(2^x+1) = x);
true
gap> ForAll([2..100], x -> Log2Int(-(2^x)) = x and
> Log2Int(-(2^x)-1) = x and
> Log2Int(-(2^x)+1) = x-1);
true
# corner cases
gap> Log2Int( 2^28 );
28
gap> Log2Int( -2^28 );
28
gap> Log2Int( 2^60 );
60
gap> Log2Int( -2^60 );
60
#
# string to integer conversion
#
gap> Int("0");
0
gap> Int("00");
0
gap> Int("01");
1
gap> Int("-01");
-1
gap> Int("100000000");
100000000
gap> Int("100000001");
100000001
gap> Int("-100000001");
-100000001
gap> Int("-100000000");
-100000000
gap> Int("123456789012345678901234567890");
123456789012345678901234567890
gap> Int("-123456789012345678901234567890");
-123456789012345678901234567890
gap> Int("");
0
gap> Int("-");
0
gap> Int("--");
fail
gap> Int("--0");
fail
gap> Int("+");
fail
gap> Int("+0");
fail
gap> Int("a");
fail
gap> Int("0 ");
fail
gap> Int(" 0");
fail
gap> Int("123456789123456789123456789123456789+1");
fail
gap> Int("A");
fail
gap> Int(['-', '1', '2', '3']);
-123
gap> Int(['1', '2', '\000', '3']);
fail
#
# integer to string conversion
#
gap> List(data, String);
[ "-100000000000000000001", "-100000000000000000000", "-10000", "-1", "0",
"1", "10000", "100000000000000000000", "100000000000000000001" ]
gap> String(10^2345) = Concatenation("1",ListWithIdenticalEntries(2345, '0'));
true
gap> String(-10^2345) = Concatenation("-1",ListWithIdenticalEntries(2345, '0'));
true
gap> STRING_INT(2^(64*1050));
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525050175624173191599128922840707048206244022665092730405123768559338458675182\
099416656451307720455955070040885306758408727902737461036072641516781109520794\
507036853608427897270817115382463919914281932999736524909005967876617990006729\
101181131806363220028599659778348171962605675653373274975893542817861600953231\
252324152953051474706804569960344757181235044174072497590120075198338233354191\
361648160768446417693467701334934747346039043646173516003421447844305336787190\
464469545957159075638994220608388396797594503810865249016738800890758262107605\
836403279617155586957440055347218377885958299749868950516157841619379649500562\
933972813356695532993304933500534044404566417076404984911048977856371003030641\
894246070142671842147610033177876614760331955660521839194430538871760201158299\
552589648431513664621868286669011154781741928754002435841244478979800030061982\
000808437628285375706683567840838443251289388892723568762941893118852489525428\
628251855165854138742268450537715753828943881263359217794580701881422153033953\
702382376225884713342595200061015331894249850790492503589957204738992129671231\
087129376234493149614871272025255929545428613203045429549880764425905365023624\
643660529727927322436421292635085224351861927144511471701114751938634359789137\
606410097806054525755691534167732940399075291334478890472473956392771617667321\
758631652229596567485774758291985339523540833762524747359698614478695609653040\
796120671293365133765885109666948031679624569135500649560641655934818588841154\
649506135219178855519572577936270442565001652279648948804823223796267452700598\
796593185054103967277906092371176795868030052057812732338087911595397862829573\
736309938934716740828666812699761855846777479766008154011376877088294162132623\
179815414912784227117818670956194195108056539836355241038628735135087870760928\
442559294364872096891526109903897875472740768999652314155371499722161322814566\
557264364907370862762525992457349009153270938100745586736497958308782267418733\
935246203095869966256390100358242731445080378636431459452908399717935069499521\
690983388979179145884187755733084200789835331119507100627639557107627871261529\
879242349612508748825664710411040842049377536164384233624681240879881015634086\
581627733464948817122577130042937085828730931030539731358775462226186271004021\
921602051206410833294106516412412224927039055722049943821715902449488344590998\
372561851557736593496185667243743138539622865137297522714623988063679374434333\
043212644256310766508806470547630696859572759762956470857184976722693245835723\
953631635925765015465470960121525139930118355061405240095695214478686152763696\
458381012040436713717847441373878992443750202425721317922485282855600755950012\
694868859981748182733043463404121437494760145844055905688381975686244812453732\
18157000453169824579899621376"
#
# RandomIntegerMT
#
gap> mysource := RandomSource(IsMersenneTwister, 42);;
gap> Random(mysource, 1, 2^20);
732950
gap> Random(mysource, 1, 2^40);
103724225096
gap> Random(mysource, 1, 2^80);
1071135285340982180054653
gap> RandomIntegerMT(fail, 1);
Error, RandomIntegerMT: <mtstr> must be a string (not the value 'fail')
gap> RandomIntegerMT("abc", 1);
Error, RandomIntegerMT: <mtstr> must be a string with at least 2500 characters
gap> RandomIntegerMT(mysource!.state, fail);
Error, RandomIntegerMT: <nrbits> must be a non-negative small integer (not the\
value 'fail')
#
# PrintInt
#
gap> for n in data do PrintObj(n); Print("\n"); od;
-100000000000000000001
-100000000000000000000
-10000
-1
0
1
10000
100000000000000000000
100000000000000000001
gap> Print(2^(64*1050), "\n");
164326883769020413733037175036750820834749677120708465325733286447124049838105\
189566407182435681174611363666903740686515184013823035067935241815636201732423\
056101003936633312198340062587975299155349963536138509122291615623302286816979\
431192334838293810345938407187189182131415855344538627014836323973615616275874\
551194116727875805708488546074947314816303386125000205325602745827110569326964\
546096605325745593681786346264019634334321319216921206034086271197556631669877\
123307235795052581387681671330910869806956328443196540735207896969831833331752\
509239464663893091463064467691031225567947499267348361811875589880987633714391\
377083732730574142699607769072424861957005067893062236517437000881328893035623\
589824257055886066984149663147620680645810096051940548433158413634136433083251\
717669215789690701323583777661028631659994503975532974307029191799551671332844\
986821852982371042719822901986744453433233917310025128730706673849226834074948\
373972654924490390643260034084145460132528646466373005866058136230963041462571\
366974476450823130983988796748258330305222046279967460890251575009744241916504\
752897541187685802641067038567640695695190264035781084289311929871639745426539\
099275420870552661756304730788198969128540003619730645216000558622501361060975\
922701569908533863633830909445630541764820170667254762497109880880481030858637\
307713666886536325529766144110338630233612264778638118214746212579284938192277\
154053325915846267023598986152825713414748358462949848367605926612482605322779\
208400820537073350290451953645109697800016938777403425589537247777527425103695\
565875198520624218787332216387043919206182706178151271009691083341678584762324\
371779288983563618238581012086246309016045282457992173599844432465680068585566\
056723045220619775226515062815701241231043061361907731142819310046497556137084\
973140722701527620338820031550997434142845001962397248729186828158341759934693\
118369636013350458850770906280968410312944448994611113747435964825608781483282\
655550777462621758016867445594217543128749839194031216608490598093509443693708\
111190031859330293998271997533868458448385236292080055236016110858874335296155\
035545864255046703941471508524077089082635498009075224739724759602363941515587\
822505372258936096574771779072070545330109662853246124245993976054726581622058\
493079009813422035721064129331732656889035515681119786520101132876923407889750\
919379880679691219867086643580120205901733738484279351295033864300058297487514\
444806928659925919270295345903616728553776703117034170898164694747612518361606\
491863490965074373346673236904951461533067041448980419457034526785836414933240\
103673756838640217363683201553347411931628209479870112525562827520962673170664\
203063656745480153500762421530947090964849895182479090513835697753334489230300\
454860027963245122579064260159518267860640914528961058131240410452474642140185\
911056636987965673332488834685650567539779734461235096967107106926330778248782\
893251703320191801146996863516822753652265863968097297820915447818880191501615\
052818274142622875924016175130106798632347377972939517614475002972499287465044\
961845657639660482751939722895448730128979063758192667864120105624208568200757\
685047135071762052798622724727343363412351393335218309185869625742046786708190\
919910685805051209816390757150374572541621648115509953333655655393114224988310\
524466349780043729997721940558172598235696785541277333196222677716540041167963\
233802695535362074133996368309233833238342788691627524874821273989033924297710\
036180702633041702593770764662210130120453714472309577105139528176114522504488\
874381647974633872068761575930834317580996045758930704230035255799797470137599\
033989839188019743852907781945117109800063009587450856630188636938603010758552\
534009632222498890557719257422859042174955764736798603271530469048159655789394\
819327180254903497066301734740246150595578336432319772069960525336675285672613\
268926838961534508137689314741119707736319964836843923205109665668325608907203\
153974944049490061297711497369306428543367744185281983262190618635576442805717\
807238930901277565544457273436300470978296111667437061064721432027113284789366\
405121630177725002727532872153537219711156904853017234855416450605426012849582\
739321172167195611208879976396751132185600538660773129192364226811739123395163\
530965557445677856159974190773165049587580425921840807124965508045214947759835\
523262151197056859619958845346458845839176068647095803197774754112655589006463\
596394338434691989671232937853536967982258563919274578951157572473714078411058\
870741720617299697357566081252160642927804820731840302037191779131861395488091\
455545163565855973696048222268435880415754170433459945880697932129038450802669\
693148706269178403831001320364375880601650598542464569186866583874317026198649\
103291885755347341686275185046692345858621355851946545615935501701280773459882\
772584863437913585811805229745858774145310938831076248152005402976642204043766\
829426397540515322742540318839716471619503601093389100475699438099103359362676\
474572936008337269304960644101631484819308659321494960959991614185255535456061\
838524800401769112986322138805393627443398061068660192511698372514815471181229\
591521938737142265095524948915771025992728119747750833679921077891786325569231\
579105807341442587144006322764442804925565207295056562928976879270882379966248\
236065780151661750508331721056915511897739600623805957111487917115483006601602\
534328815966222595505479264822920414190322384242174728987985673061155814823074\
417295708771213012567540223108379493648240851472630635992358110640101288344365\
103433189269394931204558848448793523749252366089338513571938573752327377299947\
651107861263415501447490821348636629004981208486544010636281625185969708591339\
340278759369692432329460376466491603449048550407276115976724263334104533010162\
068514474727060821941542067465600329089632202045355563098526256314203304006298\
440307792299152963245215914741179420297437350887680873989020714012250131617643\
943765728526538265506497030408069842814327847802760824539527012469159249968209\
440386157521106724825097151372591363883133349972121099516818798877576349652345\
092365608958446773034674502303503195895795521174870757634902085527933492840579\
286298066752882414465002598523465610140043634126992547913385201466840786279096\
138331476448555044597417502274416449026736524812642749847759970889053313692035\
861889467422171576860206133644291829796772006934032402537975326702505846955083\
634652203230618798935639955145043404093252838918169143944781096088613163109166\
312993726792003109579826861413407806472157247125863926006063363055412711102744\
835973698965065335987429084860341248788708261570401650009329675362478163440355\
584774989649043882327655088046404311426169490323131403037525868691322726430172\
525566993527184696640287132959276787043203061671611839315308912845697776779968\
040708568653690796727071234180737598218799951534234543378399979752798068419391\
685695639901293819346565671477580868238380412553762979132983506922946179500973\
072372424091529904207804929101962778315249964580860051623143057650828636903087\
650623239143936006261150196496039481125561859063003657456016876692219912939303\
649934354778682344087738334907714597132022551016198550077885091046612827158005\
453598886187110398597200226377804497269029390956581922772703957491877283521833\
965234557695713460434691524624004641891116966778782086446668382240671501784352\
373238026648784964313795279800042535828939669809858825678066470949394066866368\
262586823870575772795715659871488688784571302049765248202412089169503930888323\
490599966649336723180433651341659104751497601669850613850766771038237652690907\
418137856980444540467157304473112552153874924888132582078641299615430186549315\
483324262224728456537845375403309003522563763238477354542543592550315529517156\
814419992899751548087671397876713920287611278978620445452464320532087971300959\
077998059996490901450696056315029537312856613776672169545919311899621675022695\
254885190205431980383273313913364812238498968746386179758801519705215897374177\
515178886417271457708636987219068491272550942476104795618739636387669695044160\
443711597588785736901437166726709284506807205126241130231559983686986674496963\
341026521216910639932756913331939421399194056926049943917121348122841258445769\
119538929447744127511497195327108109429829422327465656942599007741240566424714\
476043257543229725866525139953103146571521055166983319403113055836697413346040\
732276457682366848413125745194735702656462473855274163364200920654288362133869\
263000408899578445934530470648567304977734981591427555431589431081364832921537\
320095286343985093254519754084126368354959325884076339776609628654646729053610\
754938871175807512231953221057185432548282454336951212492694671240112985207041\
278870849595125215446466044859452999934873724205936454142547145100218314709733\
566550109948420360572729361392113710992402646954417984933510209670249498431849\
100807846135497641988407744753491370752965599200990864318146914710140179218983\
842258654894637153469153812278058950298522443374144945961403317352489332819017\
362089054376506785537097934309720430913246306232090568916492681494152379040212\
467592300327708797959539778376158850198335903236688185013875280528527762909990\
383511389348218424714448735778715718507269113416719058132140454167547944285167\
508562348694369720572499420547037166998472329319668865080258418737293784628369\
116396967443353954193339076191448971308916427959775389107197691219066540903585\
843396201421682561264380473222295169713288050744782428013579400569861165347801\
740958755792598216905044291792575500887961296663574454157657015014794662012832\
002763987953996864811130510341906634257143039133477910171960514510326773417300\
473350876261727637054579584094615225038191328108855914857992322565158030357029\
948694667751167287226128755830073291955994960608682299840800136371303082193744\
102984706514935794768294085837145063769215606265902771574069545414198048071682\
323953018978863908610461546156004162901986384823772103911557900478950149990147\
505642664635313842039606239287972229581707268402575983318389255488573245848455\
672051513804995262247510365565343775075398971487108014908523800542298273812311\
891009565647082789733497736679341679613362798203424057176142401292361689914382\
600656643076243695475065503125124713563303612336242545180794074700756287428778\
918043850756404664697029692492317678961386639599904615136222047774182637237883\
582038612300586126006812687129636338394989180638835422270047472081947444527380\
590474160031930624241375064121019415304653566732088972342211810949500111959095\
011659481725841299033883029036542767843383377165673198140517988235663348811804\
380190747862108749889646024182989130466206454334313388265808747847693185414083\
846907193092393416556105411497066769114689573399652247156075189966590116140386\
024299086892962087455129066805108952049644614913789438084914079647893567959676\
900193774643239455728954737417343518742297186938749892875005282590763806410508\
343440551321139153691492802683069617087386978139831173995574941770196218409212\
086200402559581963745903809226209167667567320315027041696858734517597712537031\
481306983739528080579374622370200807888569632824561038555301130064947884319827\
573391049544447546654993147085501593897075354256479398825280710704693246089049\
544233251458129335315549062693514323110980809159958128694466940307392240769241\
118169392673506207828271638716787304653644293263205010248938261256488472601055\
821780090511720169053009134349232423810588661841217966450172198109392679232730\
595007818198254921826376858500505686299849027959021412012184814724881786371951\
668164206022905059852364989992302711809736691284928294640795189921639147009924\
650282186949665024092458113495124905683278972982377212979145794201591378761625\
488243059789755743738286227419164973798162967975446589964441830994494819573900\
817991663859116404727330000723620324426994450712204462960582580194486365653844\
270278424077064126226480523602393447511372932688789465431452363631837605612532\
972920400444096699455687738764897304770578984415450917343085944682174835992976\
032109663793736157173096599930006009196502279311229593159433146219733056913413\
467552179773553196794161284567014689331940232843120310294737419642804285393711\
423142039441538552804116540852527085831052031390968590250230892958728434286639\
030114504755247612648263356452731995160976995390550000062104315649414537804526\
039500531080784141890907315815476835485667012616479169122239684554640700775183\
357322802537034848032140383973736546694162487667323503483745793562451900943135\
221737570340620888370840573789605697469492243553016237126377167013785598471254\
099282306926679184768973077096472867165675822972745199941777989313088851497198\
503083648676778655693971659405157247801978509297794003926428010451337912177800\
984617926928081340343106635444992606093461930025856987540144386957289542712917\
708404919440142785591004965606390335022886056059514560248768675087413439764323\
447877144011250556103187246077786973347672777353815550684934380536899468069334\
104716136402315974341243242034684301593178102332747368012628020715027261952950\
717174690818654516595206965149573075967691524795498691231398313987841940244411\
503309410056436795336137181380745781817674535875910686428675852120624771185051\
339088218274761410431459347482785946119131434469998295202470762186075662087296\
135360849993093629832279709005000897185086161787695581730439201011882290665996\
324005537416847036797238552005884909301698377453218545651175892922885429563912\
177380161406881091927909681169361647145140516691522268369182236591244089063250\
722219544526066675698553828609740161485112326877077404225835880828594404528752\
645291083854562674855334557715603138349329916462980744602616541860699285164984\
798192895527446260336716079018511929124606869940323998112697771772676180970532\
359193310838696649412166913286939973669672830683499051070065499204851250690065\
504487276965856911392035288639223599696403047515257815189448945589325275527098\
679774285846725549341313740869610743497323830419065537721433043215944536924524\
801036593430833634879594218476429290634411658113636300276587779353098788310947\
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#
# test InversModInt / INVMODINT
#
gap> for m in [1..100] do
> for b in [1..100] do
> i := INVMODINT(b,m);
> g := GcdInt(b,m);
> if g = 1 then
> Assert(0, i in [0..m-1]);
> Assert(0, (i*b-1) mod m = 0); # formulated this way to support m=1
> Assert(0, i = INVMODINT(b,-m));
> else
> Assert(0, i = fail);
> fi;
> od;
> od;
#
gap> INVMODINT(1,0);
Error, INVMODINT: <mod> must be a nonzero integer (not the integer 0)
gap> INVMODINT(2,10);
fail
gap> INVMODINT(2,-10);
fail
gap> INVMODINT(2,2^80);
fail
gap> INVMODINT(2,1);
0
gap> INVMODINT(2,-1);
0
#
# test PowerModInt
#
gap> for m in [1..100] do
> for b in [-30..30] do
> for e in [0..10] do
> Assert(0, PowerModInt(b,e,m)=(b^e mod m));
> od;
> od;
> od;
#
gap> PowerModInt(1,1,0);
Error, PowerModInt: <mod> must be a nonzero integer (not the integer 0)
gap> PowerModInt(0,-1,5);
Error, PowerModInt: negative <exp> but <base> is not invertible modulo <mod>
gap> PowerModInt(2,-1,5);
3
gap> PowerModInt(2,-1,10);
Error, PowerModInt: negative <exp> but <base> is not invertible modulo <mod>
# compare C and GAP implementation
gap> POWERMODINT_GAP := function ( r, e, m )
> local pow, f;
>
> # handle special cases
> if m = 1 then
> return 0;
> elif e = 0 then
> return 1;
> fi;
>
> # reduce `r' initially
> r := r mod m;
>
> # if `e' is negative then invert `r' modulo `m' with Euclids algorithm
> if e < 0 then
> r := 1/r mod m;
> e := -e;
> fi;
>
> # now use the repeated squaring method (right-to-left)
> pow := 1;
> f := 2 ^ (LogInt( e, 2 ) + 1);
> while 1 < f do
> pow := (pow * pow) mod m;
> f := QuoInt( f, 2 );
> if f <= e then
> pow := (pow * r) mod m;
> e := e - f;
> fi;
> od;
>
> # return the power
> return pow;
> end;;
gap> for m in [1..100] do
> for b in [-30..30] do
> for e in [0..30] do
> Assert(0, POWERMODINT_GAP(b,e,m)=PowerModInt(b,e,m));
> if GcdInt(m,b) = 1 then
> Assert(0, POWERMODINT_GAP(b,-e,m)=PowerModInt(b,-e,m));
> fi;
> od;
> od;
> od;
#
# test PVALUATION_INT
#
gap> checkPValuationInt:=function(n,p)
> local k, m;
> k:=PVALUATION_INT(n,p);
> if n = 0 or p = 1 or p = -1 then return k = 0; fi;
> m:=n/p^k;
> return IsInt(m) and (m mod p) <> 0;
> end;;
gap> ps := [-2^60-1,-2^60,-2^28-1,-2^28];;
gap> Append(ps, [-6..-1]);
gap> Append(ps, [1..20]);
gap> Append(ps, [250..260]);
gap> Append(ps, [2^28-1,2^28,2^60-1,2^60]);;
gap> SetX([-1000 .. 1000], ps, checkPValuationInt);
[ true ]
gap> SetX([-1000 .. 1000], ps, {n,p}->checkPValuationInt(n+2^100,p));
[ true ]
gap> SetX([-1000 .. 1000], ps, {n,p}->checkPValuationInt(n-2^100,p));
[ true ]
#
gap> p:=2^255-19;; # big prime
gap> ForAll([1..30], k-> PVALUATION_INT((p+1)^k,2)=k);
true
gap> ForAll([1..30], k-> PVALUATION_INT((p+1)^k,p)=0);
true
gap> ForAll([1..30], k-> PVALUATION_INT(p^k+1,2)=1);
true
gap> ForAll([1..30], k-> PVALUATION_INT(p^k+1,p)=0);
true
#
gap> SetX(data, dataNonZero, checkPValuationInt);
[ true ]
gap> List([2..6], p->List(data, n->PVALUATION_INT(n,p)));
--> --------------------
--> maximum size reached
--> --------------------
[ Dauer der Verarbeitung: 0.40 Sekunden
(vorverarbeitet)
]
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